Question 1136404
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Since all the measurements are whole numbers, we can easily solve the problem by guessing, knowing that the side lengths of right triangles KMN and KML are Pythagorean triples.<br>
Since MN=13, KN is almost certainly either 5 or 12.  Trying KN=5 makes MK=12 and LM=9; and that makes triangle KML a Pythagorean triple 9, 12, and 15.<br>
So triangle KMN is a 5-12-13 triangle with area (5*12)/2 = 30; triangle KML is a 9-12-15 triangle with area (9*12)/2 = 54.<br>
Then the area of KLMN is 30+54 = 84.<br>
The problem can be solved formally using the Pythagorean Theorem.<br>
Let KN = x; then LM = x+4.  And let KM = y.  Then<br>
{{{x^2+y^2 = 13^2 = 169}}}
{{{(x+4)^2+y^2 = x^2+8x+16+y^2 = 225}}}
{{{8x+16 = 56}}}
{{{8x = 40}}}
{{{x = 5}}}<br>
Then the rest of the path to the answer to the question is the same as above.